Evolution equations with nonlocal initial conditions and superlinear growth

TL;DR

This paper investigates the existence of solutions for nonlinear parabolic PDEs with nonlocal initial conditions, including superlinear growth nonlinearities, using an abstract approach and continuation principles.

Contribution

It introduces a novel combination of approximation and continuation methods to handle superlinear nonlinearities without Lipschitz or compactness assumptions.

Findings

Proves global existence of solutions for a broad class of nonlocal parabolic problems.

Establishes the existence of at least one periodic solution in the periodic case.

Handles superlinear growth nonlinearities without requiring Lipschitz continuity.

Abstract

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case we are able to prove the existence of at least one periodic solution on the half line.